Axel von Engeln 1,2, Gerald Nedoluha 3

Retrieval of temperature, water vapor, and a reference pressure from radio occultation refractivity and bending angle measurements using a 1D Var approach: A simulation study Axel von Engeln 1,2, Gerald Nedoluha 3 Abstract. A 1D Var assimilation study of simulated radio occultation measurements is presented. High resolution European Center for Medium Range Weather Forecasts global atmospheric fields are read by a state-of-the-art radio occultation simulator. The simulator provides quasi-realistic bending angle and refractivity profiles. Both profiles have been used in the assimilation process to assess the advantages and disadvantages of one over the other when retrieving temperature and water vapor profiles. Also, 2 different a priori (background) fields are used to assess possible impacts of the a priori quality. An optimization with respect to the convergence was performed first, where initialization of the Levenberg Marquardt method closer to a Newtonian iteration was generally found to be advantageous, although lower tropospheric retrieval might benefit from a steepest descent approach. Retrievals using quasi-realistic measurements showed on average errors of below.2 K (. K) for an altitude range of 11 km to 2 km (9 km to 2 km) using an optimistic (moderate) a priori. Especially retrievals in the lower troposphere are mainly affected by the inability of the 1D Var algorithm to effectively separate the impact of water vapor and temperature. Generally, lower tropospheric retrievals provide mainly information about water vapor in the tropics and at mid-latitude, while polar retrievals are mainly providing temperature information. Overall, assimilation of refractivity profiles shows several advantages, among them a lower sensitivity to erroneous water vapor retrievals, better temperature retrieval capabilities in the upper stratosphere, and faster CPU executions. 1. Introduction Global Navigation Satellite Systems (GNSS), such as the American Global Positioning System (GPS), the Russian Global Navigation Satellite System (GLONASS), or the future European Galileo system, provide a continuous source of signals at radio frequency. The frequency of these signals is within a range where the impact of clouds can generally be neglected, hence they provide an interesting field for atmospheric remote sensing. GNSS receivers can either be ground-based or space-based, on the ground they allow to derive information about the integrated precipitable water vapor [Bevis et al., 1992], while air or space born re- 1 George Mason University Fellow at Naval Research Laboratory, Monterey, CA, USA 2 Institute of Environmental Physics, University of Bremen, Bremen, Germany 3 Naval Research Laboratory, Washington, D.C., USA ceivers provide profile information on temperature and water vapor [Fischbach, 196; Kursinski et al., 1997; Zuffada et al., 1999]. The principle geometry of the radio occultation measurement for a space borne GNSS receiver in a Low Earth Orbit (LEO) detecting signals from GNSS satellites in an occultation geometry is shown in Figure 1. The measurement can be converted to a profile of bending angles α, where each α can be expressed as a function of the impact parameter a or the radius to the tangent point r t, as given in Figure 1. The impact parameter is defined as the distance between the Earth s center and the perpendicular on the asymptotic ray path. The radius to the tangent point is defined as the point along the path were the distance to the Earth s surface is minimized. The bending angle profile can be converted to a refractivity profile by applying an Abel transform [Fjeldbo et al., 1971]. The situation given in Figure 1 holds only if local spheri- 1

2 VON ENGELN AND NEDOLUHA r t α GNSS a a LEO Earth Atmosphere Figure 1. Radio occultation geometry in a spherically symmetric medium, the impact parameter a, the bending angle α, and the radius to the tangent point r t are indicated. cal symmetry is reasonably approximated. Deviations from local spherical symmetry are introduced by atmospheric inhomogeneities and the elliptical shape of the Earth. The elliptical shape of the Earth can be compensated for by the introduction of a different Earth s center and a radius of curvature, depending on longitude and latitude. This radius is a fit to the actual shape of the Earth at the occultation event, it is larger than the Earth s radius at the poles and smaller than the Earth s radius at the equator. For a more thorough error discussion please refer to Zou et al. [2]. The influence of deviations from a spherical symmetric atmosphere can not be corrected this easily and will in general introduce errors in the retrieved atmospheric profiles, e.g. [Healy, 1]. A first proof-of-concept mission for radio occultation was the GPS Meteorology (GPS/MET) experiment, led by the University Corporation for Atmospheric Research (UCAR) in Boulder, U.S.A. [Ware et al., 1996]. Several thousand GPS/MET measurements have been compared with correlative data sets and a statistical agreement within 1 K mean temperature for an altitude range of 1 km to 4 km was found [Rocken et al., 1997]. Most of the early retrievals of temperature and water vapor profiles from radio occultation measurements were focusing on a direct retrieval approach, thus no a priori (also called first guess, or background) information was incorporated. This allowed for the determination of a dry temperature profile where the amount of water vapor was insignificant. At lower tropospheric altitudes either temperature or water vapor can be determined assuming that the other quantity is well known. More recently the process of bending angle or refractivity assimilation by variational systems has been addressed. These systems are defined by the number of dimensions considered, they span the range from 1D Var to 4D Var, e.g. [Palmer et al., ; Kuo et al., ; Liu et al., 1; Poli et al., 2], and allow for the simultaneous determination of temperature, and water vapor profiles. No comprehensive comparison of bending angle profile assimilation versus refractivity profile assimilation has been made so far. Furthermore, no quasi-realistic simulation study using high resolution global atmospheric fields has been conducted to quantify the potential of a variational approach to radio occultation processing. Within this study we shall use simulated radio occultation measurements made from a single LEO satellite for one particular day in 1 (May 19, 1). This day lies within the time frame of a Naval Research Laboratory assimilation study, focusing on the impact of radio occultation data on a Numerical Weather Prediction (NWP) model. One LEO satellite, using only signals of the GPS constellation, would be able to detect about 2 setting events and 2 rising events per day [Kursinski et al., 1997], a fully installed GLONASS or Galileo system would double the number of soundings. In total GPS occultation events were found for our simulated LEO satellite, and a subset of 1, chosen to provide a wide range of latitude measurements, was used in this study. Measurements were simulated using a ray tracing or wave optics propagator model and a high resolution European Center for Medium Range Weather Forecasts (ECMWF) dataset. A 1D Var retrieval method was used to derive temperature and water vapor profiles, and a reference

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS 3 pressure. The primary purpose of this study is to provide quantitative assessments of the 1D Var retrieval capabilities of temperature and water vapor profiles from radio occultation measurements and to determine whether the assimilation of bending angle or refractivity profiles is more favorable. Our paper is structured as follow: Section 2, 3: introduction of the forward and retrieval model applied; Section 4: convergence of the retrieval process; Section : presentation of quasi-realistic retrieval results; Section 6: impact of assumed a priori errors; Section 7: impact of assumed a priori correlations; Section 8: impact of assumed a measurement errors; Section 9: impact of assumed a correlations in the refractivity measurement; Section 11: conclusion. 2. Forward Model Setup Two forward models have been used in this study, a 3D End-to-end GNSS Occultation Performance Simulator (EGOPS) to produce quasi-realistic measurements, and a 1D forward model for the retrieval processing. Within the simulator tool, ECMWF atmospheric analysis fields of May 19, 1 with 4 time steps, 6 vertical levels, and a T11 horizontal resolution were used [Miller, 1999; Teixeira, 1999; Jakob et al., ]. The vertical resolution of these fields gradually increases from m at the surface to about 2 m at 1 km altitude, and about 1 km to 3 km in the stratosphere. The horizontal resolution is about.31. The upper limit of these fields is around 6 km. We use the MSIS atmosphere for altitudes above 6 km [Hedin, 1987, 1991]. 2.1. 1D Forward Model The forward model used for the retrieval processing is 1-dimensional, and therefore neglects the effects of horizontal variations in the atmosphere. It has already been used for a combination study of a radio occultation with a passive microwave instrument [von Engeln et al., 1] and a radio occultation sensitivity study [von Engeln et al., doi:.29/2jd298, 3]. The principle equations of the radio occultation 1D forward model are given in Kursinski et al. [1997]. Refractivity N at each atmospheric level is approximated using the formula from Smith and Weintraub [193]: N = 77.6 p T + 3.73E e T 2 (1) with T atmospheric temperature [K], p atmospheric pressure [hpa], and e partial water vapor pressure [hpa]. The refractive index n is calculated at each level from N as: n = 1 + N 6 (2) The calculation of the bending angle α as a function of refractive index follows an Abelian integral equation. The forward Abel integral is given by: r= 1 α(a) = 2a r=r t (n r)2 a 2 t d ln(n) dr dr (3) where the integration is performed through all altitudes r down to the tangent point r t, and the factor 2 results from the assumed symmetric atmosphere. The impact parameter at r t is a t = n t r t. Within the radio occultation processing, the transform of this equation is used to calculate the refractivity profile from the bending angle measurements. The internal resolution for the evaluation of Eq. 1 and Eq. 3 in the 1D forward model is set to. km to capture the high resolution ECMWF fields. Input profiles are interpolated to this resolution using a polynomial function. Table 1 summarizes the range and the sampling (reflecting the resolution) of the measurement. Bending angles and refractivity profiles are transformed from the internal forward resolution to the given measurement resolution by averaging over the corresponding vertical intervals. Fresnel diffraction will generally lead to a resolution of about. km in the lower atmosphere, if no diffraction correction is performed. The canonical transform, as discussed by Gorbunov [2a] can theoretical improve the resolution of radio occultation data to within 3 m, but errors of the measurement will limit the resolution practically to about m to m [Ao et al., 2]. Hence the chosen resolution reflects a compromise between this possible high resolution and practical applications, e.g. the assimilation of radio occultation data into a NWP model. Global NWP models currently provide a vertical resolutions of up to 2 m at an altitude of 1 km, hence this resolution exceeds NWP model resolutions at altitudes of about 1 km [Teixeira, 1999]. Table 1 lists as well the assumed measurement errors on the bending angle and refractivity profiles, which are inbetween the proof-of-concept GPS/MET instrument and expected accuracies of modern receivers [ESA/EUMETSAT, 1998]. Refractivity errors are approximated by assuming the same signal-to-noise ratio as given for bending angle measurements. This is a crude assumption of refractivity errors which are usually assumed to be about 1 % at the surface, and linearly decreasing to about.2 % at km tangent altitude, and staying constant above [Healy and Eyre, ]. The impact of this assumption is addressed within this study by varying the measurement errors. Useful determination of α and hence N above 6 km is not possible, due to the poor signal-to-noise ratio of mesospheric radio occultation data.

4 VON ENGELN AND NEDOLUHA Table 1. Sampling grid used for the simulated measurements, along with the corresponding bending angle () and refractivity () errors assumed. Height Sampling dz Error Error [km] [km] [µrad] [1] z 2.2 4.. 2 z 4. 2.8.36 4 z 6 1. 2..27 2.2. 3D Ray Tracer The EGOPS simulator tool [Kirchengast, 1998; Kirchengast et al., 2] was used to simulate bending angle and refractivity measurements. EGOPS reads in orbital parameters for the GNSS and the LEO satellites and predicts in a first step approximated occultation locations. Our scenario is based on the future EUMETSAT Polar System satellite (also called MetOp 1), the first operational radio occultation instrument planned for launch in, with an orbit altitude of 84 km and an orbit inclination of 98.7. In the second step, EGOPS reads in atmospheric fields and uses a ray tracer or wave optics forward model propagator with sub-millimeter accuracy to calculate the amplitude and phase of the simulated ray path, along with its precise occultation location. The ray tracer includes ionospheric effects simulated by a 3D ionosphere with a solar activity index of 13 (.7 cm flux). The tracing terminates when one of the rays hits the Earth s surface, thus not all occultations reach down to the surface. Wave optics calculations are performed without the ionospheric influence, the calculation uses a similar termination criteria as the ray tracer. Almost all 1 simulated ray tracing occultations reach tangent altitudes km, but only about reach tangent altitudes < 1 km. Wave optics calculations terminate slightly lower in the atmosphere, leading to about 3 occultations that reach altitudes < 1 km. The third step of an EGOPS simulation consists of the observation system modeling. This denotes the superposition of all relevant physical and technical influences of the observation system onto the output of the second step, and on the ideal orbit data (GNSS and LEO positions and velocities). Relevant influences include effects of precise orbits, simulation of the receiving system, antenna specifications, receiver noise, multipath impacts, and the differencing treatment and clock modeling. All error characteristics follow those of modern receivers [ESA/EUMETSAT, 1998]. The output of this third step represents the raw measurement of a LEO GNSS receiver. The last step of an EGOPS calculation processes the simulated phase and amplitude data (supplemented by the necessary geometrical information) via Doppler shifts and bending angles down to quasi-vertical atmospheric profiles of refractivity, density, pressure, temperature, and humidity. Within this study, bending angle and refractivity profiles were used as quasi-realistic measurements. Both profiles were pre-processed before entering the retrieval processing to convert them into the format used within the 1D forward model, this was performed in a similar manner as described above. 2.3. Diffraction Effects Neither ray tracing nor Abel integral based forward models consider explicitly the effects of Fresnel diffraction, which will limit the vertical resolution of the measurement to approximately the diameter of the first Fresnel zone (if no diffraction correction is performed). The size of the first Fresnel Zone is usually around. km in the lower atmosphere, and increases to more than 1 km in the upper stratosphere [Kursinski et al., 1997]. Simulations generated from wave optics calculations are used to address the impact of diffraction. 3. Retrieval Model The inverse model calculates the most likely solution ˆx of the true atmospheric state x. A Bayesian solution based on a linear inverse problem leads to the definition of a cost function; the most likely solution is found by minimizing the cost function. Since the retrieval of temperature and water vapor profiles from radio occultation measurements can be non-linear, an iterative approach is applied. The retrieval algorithm used here is based on the 1D Var or Optimal Estimation Method [Rodgers, ]. It uses a priori knowledge on the state of the atmosphere to stabilize the solution. The iterative formula to calculate ˆx for the iteration n + 1

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS is given as: ˆx n+1 = x + G n [(y y n ) K n (x ˆx n )] (4) where x is the a priori vector from which the iteration starts, G n the n-th iteration of the gain matrix, y represents the bending angle measurement, y n the forward model output, and K n the Jacobian. The Jacobian matrices K n and G n are defined as: K n F (x) x x=xn G n I(y) y y=yn=f (x n) where F (x) is the forward model discussed in the previous section, which creates simulated measurements for any given state x. For this study the Jacobian matrix K n is calculated by perturbing the corresponding retrieval parameter x of F (x). For the given formulation the matrix G n can be calculated from: G n = (S 1 + K T n S 1 y K n) 1 K T n S 1 y (6) with the a priori error covariance matrix S, the error covariance matrix of the measurement S y, and K T n denoting the transpose matrix of K n. The 1D Var allows the calculation of two statistical errors, each of which contributes to the total error of the retrieval. Generally one can write the total retrieval error covariance matrix S as: S = S S + S M (7) where S S is the smoothing error covariance matrix, and S M is the measurement error covariance matrix. The error as defined in Eq. 7 is not used within this study. Instead we characterize the retrieval quality by calculating the mean absolute error E for temperature (T) and water vapor (WV) at each retrieval level i as follows: E T i = 1 n E WV i = 1 n () n x j ˆx j (8) j=1 n (x j ˆx j )/x j (9) j=1 where the number of available retrievals at level i is given by n and the true profile x is defined as the one following the tangent point trajectory. This method of error calculation provides a more robust statistic over standard deviations, since outliers are less weighted. Possible biases that might be either present in the a priori data or are introduced within the 1D Var processing are consequently not removed, but are discussed separately. 3.1. Retrieval Setup Within this work we distinguish between two different retrieval approaches: ideal and realistic. Ideal Retrieval: An ideal retrieval case can be defined by assuming the same forward model in the generation of a simulated measurement and within the retrieval model. The measurement is free of biases, ionospheric effects, and other observation system influences, thus only Gaussian noise is considered. Furthermore, the temperature, water vapor, and pressure profile are only defined on the actual retrieval grid, representing a 1D calculation both in the forward and the retrieval model. Thus no fine scale structure, as discussed in von Engeln et al. [doi:.29/2jd298, 3], is present. Realistic Retrieval: A realistic retrieval uses bending angles or refractivity profiles derived by the EGOPS tool as measurement vector, including error specifications used for simulation studies of modern radio occultation receivers [ESA/EUMETSAT, 1998]. Thus this retrieval approaches is based on a 3D forward model, but within the retrieval, the 1D forward model is used. Hence, a comparison of ideal and realistic retrievals allows to estimate the errors introduced by horizontal inhomogeneities, the ionosphere, and measurement errors. Both retrieval approaches use the same ˆx vector, which holds the temperature profile between km and km, the water vapor profile between km and km, and a reference pressure from which the hydrostatic atmospheric pressure profile is generated. The reference pressure is retrieved at the lowest retrieval altitude. Water vapor is expressed in volume mixing ratio (VMR) units. Since VMRs vary by more than 2 orders of magnitude on a linear scale, artificial oscillations and negative values can occur in the retrieved water vapor VMRs [Bühler, 1999]. These problems can be overcome by using the logarithm of the water vapor VMR in the retrieval vector ˆx. Although this changes the assumption on the a priori errors. It usually follow a Gaussian or normal distribution, thus the log(vmr) errors follow a lognormal distribution. The log(vmr) units also give a positive constraint on the retrieval of water vapor. Thus, log(vmr) is generally used in the 1D Var retrieval from radio occultation data [Healy and Eyre, ; Palmer et al., ; Poli et al., 2], but one should keep in mind that the log(vmr) distribution is asymmetric. The vertical retrieval grid is given in Table 2, it is chosen to match the measurement resolution as given in Table 1 in the lower atmosphere, to avoid the introduction of retrieval errors caused by a coarser retrieval grid [von Engeln et al., doi:.29/2jd298, 3]. The retrieval steps are gradually increased above km to compensate for the de-

6 VON ENGELN AND NEDOLUHA creasing signal-to-noise ratio. Retrieval above 6 km is not possible with the chosen setup, nevertheless the extension of the temperature grid up to altitudes of km assures that uncertainties in the mesospheric temperatures will be considered in the error budget at lower altitudes. Contributions to the error budget arise from the limb sounding geometry. The S matrix is generally generated with a 2. K a priori uncertainty for temperatures up to km and a linear increase up to K at km. For water vapor a 4 % uncertainty is generally assumed, and a 1 % error in the reference pressure. A NWP model short range forecast calculation can provide an a prior temperature and water vapor profile. The forecast uncertainty for temperature of an NWP model is about 2. K for the lower atmospheric layers (up to km) and increases to about 1 K at 6 km [Palmer et al., ]. Water vapor forecast is possible with an uncertainty of about 4 % [Palmer et al., ]. Hence our a priori profile uncertainties reflect current forecast capabilities of NWP models. The 1 % a priori uncertainty for the reference pressure retrieval is a conservative assumption, but sensitivity of the retrieval to this uncertainty is very low. Both, temperature and water vapor uncertainties have been varied within this study in order to assess the impact. A priori profiles themself are taken from ECMWF forecast or analysis fields. We use two different a prioris, the first one is based on a 12 h to 18 h short term forecast, denoted optimistic case. The second a prioris are based on the analysis field of the previous day (May, 18 1), denoted as moderate case, hence assuming that the temperature and water vapor profiles did not change significantly within one day. The mean tangent point position of the occultation was used to extract a 1D a priori profiles from the ECMWF fields, possible improvements in the a priori profile, as outlined in von Engeln et al. [doi:.29/2jd298, 3], have been disregarded. The a priori uncertainties as outlined above are identical for both a priori cases, assuming a conservative setting. The impact of variations in the uncertainties is discussed in a separate section. A priori profiles are in general assumed to be unbiased, with no vertical correlations. The impact of correlations in the temperature and water vapor profiles is addressed in a separate section. Generally, the a priori profiles and their error covariance matrices have to be chosen very carefully for a retrieval from real data, in order to avoid the introduction of a bias [Rodgers, ]. The measurement covariance matrix S y is generated with the errors presented in Table 1. Bending angle and refractivity profile errors are based on the same signal-to-noise ratio, to allow easier comparison. We shall assume that the bending angle measurement error covariance is statistical in nature (i.e., the measurement profile have no systematic bias) [Rieder and Kirchengast, 1]. The processing of bending angles to a refractivity profile involves the inverse of Eq. 3, which is essentially a weighted sum of the bending angle measurements. This processing introduces correlations in refractivity measurements at different altitudes [Healy and Eyre, ]. These correlations can be included in the measurement error covariance matrix, but for simplicity we will not include these correlations in the retrievals shown here, except when explicitly indicated. The iterative process in Eq. 4 is always terminated after iteration 4, since convergence was usually found within 4 iterations. Nevertheless, more iterations have been calculated to verify that the major findings of this study remain unchanged, unless otherwise indicated. A more thorough discussion on the iterative behavior of the processing is discussed below. 4. Convergence The iterative process as given in Eq. 4 is based on Newtonian iteration which is a straightforward numerical method for finding the minimum of a not too non-linear cost function. However, this iteration can perform non-optimal for more non-linear cases, and the Levenberg Marquardt method is frequently used instead [Rodgers, ]. Depending on the γ initialization value of the Levenberg Marquardt method, this method either starts with a steepest descent approach or a Newtonian iteration. Choosing the initial γ close to results in a Newtonian iteration, while high γ values result in a steepest descent approach. Within the iteration process, the initial setting is updated to optimize the solution finding, thus initializations can in general be chosen conservative with a setting closer to steepest descent approach. Although, the minimization process might be altitude dependent, since upper tropospheric, stratospheric temperatures are expected to behave more linear, while lower tropospheric retrievals should be more non-linear caused by the impact of water vapor. First ideal and realistic retrieval calculations used a conservative Levenberg Marquardt parameter setting with a fixed number of 4 iterations. Generally one terminates the iteration process based on a combination of a convergence criteria and a maximum number of iterations allowed. Thus, for computer efficiency, it is important to determine the required number of iterations in order to terminate the process of minimization finding. Calculations for a limited set of occultations showed convergence within 4 iterations. However, during processing of a larger dataset, especially midlatitude and polar retrievals from bending angles occasionally showed unrealistic water vapor results near the maxi-

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS 7 Table 2. Vertical retrieval grid. Species Height Sampling dz [km] [km] Temperature z.2 z 3. 3 z 4 1. 4 z 6 2. 6 z. Water Vapor z.2 mum altitude where the retrieval was still sensitive to water vapor. Figure 2 shows such a problematic retrieval for a polar occultation using a moderate a priori. Between about 4 km and 9 km the a priori water vapor profile, as given by the iteration, differs by less than % wrt the true profile. The iteration process does not succeed in minimizing the difference to the true profile, instead the initial % difference is four-folded with iteration 4. The temperature iterations show that the water vapor adjustment is done in order to minimize the difference between the a priori and true profile around 4 km and 9 km. The bending angle results show that the iteration process was clearly able to move toward the minimum in the cost function, and the difference of iteration 4 to the true profile is close to. Increasing the number of iterations will gradually remove the strong water vapor adjustment, but such an approach is undesirable in order to optimize the retrieval processing and minimize CPU time. Furthermore, slow convergence might fall through a set convergence criteria, thus the algorithm stops at a non-optimal solution. A different Levenberg Marquardt initialization closer to a Newtonian iteration shows better results within 4 iterations, even though the water vapor profile is still adjusted to a difference of around % around 6 km. But temperature results are much better around km to 9 km and the bias at altitudes above about km is reduced. These findings apply to both, bending angle and refractivity assimilation. Ideal retrievals with different Levenberg Marquardt initialization were performed for all investigated 1 occultations. Ideal retrievals were chosen in order to have a cost function unobscured by possible simulated measurement errors and forward model inconsistencies. The initialization of the Levenberg-Marquardt scheme was varied by a factor of between and.1. A subset of these calculations is shown in Figure 3, where the error as defined in Eq. 8 is plotted. Retrievals were performed down to the lowest possible altitude, as given either by ducting or by the surface topography. Ducting will cause rays to bend down toward the surface and the ray might be lost. Note that ducting conditions can occur in the simulated measurement and in the a priori profiles, thus even when ducting was not present in the measurement, a reduction of the retrieval range can result from the a priori. Only the bending angle calculation is affected by ducting conditions in the a priori profiles, although for this study, the retrieval range of both assimilation variables was restricted for these cases. For more information about ducting occurrence in ECMWF fields, please refer to von Engeln et al. [2]. Optimistic a priori: Temperature information is provided by the a priori with a mean absolute error of about. K to 1. K for the considered retrieval altitudes, as shown in the top left plots of temperature retrieval from bending angle or refractivity measurements (Figure 3). The mean absolute error of the water vapor a priori is within % to 3 %, except for 2 spikes at about 2 km and 3 km (shown in the top right plots of water vapor retrieval from bending angle or refractivity measurements in Figure 3). These two spikes are caused by 2 occultations, they occur in the true profiles of 2 tropical occultations, a strong vertical water vapor gradient is found at altitudes corresponding to the planetary boundary layer. Temperature retrieval from bending angle or refractivity profiles look very similar for low Levenberg Marquardt parameter settings. Settings closer to a steepest descent approach degrade the retrieval capabilities from bending angles at higher altitudes, while refractivity results show only small variability. Thus the retrieval needs more iterations to converge to the right solution for bending angle assimilation. At lower altitudes, an initial steepest descent setting seems advantageous both for bending angle and refractivity assimilation, however calculations with a higher number of iterations lead to similar results obtained with a lower set-

8 VON ENGELN AND NEDOLUHA 1 Iter Iter 1 Iter 2 Iter 3 Iter 4 -. -... Temp wrt true [K] -. -... WV wrt true [%] -. -.... wrt true [%] Figure 2. Ideal retrieval iterations with respect to true profile for occultation 1 (Occultation Location: 66.7 N, 94.6 E). Left: temperature; Middle: water vapor; Right: bending angle. ting, but convergence is very slow. Water vapor retrieval results are almost unaffected by the chosen Levenberg Marquardt setting, even though retrievals from bending angles seem to perform better at one of the spikes for high Levenberg Marquardt parameter settings. Calculations with a higher number of iterations show that convergence was not reach within 4 iterations with higher Levenberg Marquardt parameter settings. Moderate a priori: Temperature a priori information with a mean absolute error of about 1 K to 3 K is provided for the considered altitude range (bottom left plot of Figure 3). Water vapor a priori errors lie within % to 8 %, if one disregards the spikes around 2 km and 3 km (bottom right plot of Figure 3). Temperature retrieval results are similar to those obtained with the optimistic a priori. Levenberg Marquardt parameter settings closer to a steepest descent achieve better results below about 8 km, independent of the assimilation variable. Refractivity assimilation again requires less iterations in the upper troposphere, lower stratosphere for settings closer to Newtonian iteration. Water vapor retrieval results are again almost unaffected by the settings, with a small exceptions: Retrievals from bending angle measurements have a tendency to converge slower to the minimum in the cost function, as mentioned above (visible at altitudes near 7. km). Convergence was not reached within 4 iterations for the higher Levenberg Marquardt parameter settings. As was discussed in Section 3, the use of refractivity as the measurement variable introduces correlations in the measurements at different altitudes. We find that when these correlations are included in the measurements error covariance matrix the results are very similar to the ones presented, although the temperature retrieval accuracy is degraded in the upper troposphere, lower stratosphere, similar to the presented results for bending angle assimilation. All retrieval results were verified against calculations with a higher number of iterations, showing that they all converge to the same, optimal solution. These calculations showed no further improvement over the low Levenberg Marquardt parameter setting at higher altitudes. The higher settings do not converge within 4 iterations at low altitudes, more iterations are required. Convergence was reached quicker with a low setting. Hence we use a uniform setting of.1 throughout the rest of the study, since this provides fast convergence and avoids convergence problems around the upper limit for mid-latitude water vapor retrieval. The reference pressure was always retrieved at the lowest retrieval altitude, bias and mean absolute error are presented in Table 3 for the bending angle assimilation scheme. Refractivity assimilations generally yields very similar results. There is no improvement in the mean absolute error of the pressure retrieval. This is caused by incorrect temperature adjustments at the lower altitudes, which in term is caused by the water vapor uncertainty. The mean retrieval error increases with lower Levenberg Marquardt parameter settings, since the quality of the temperature retrieval de-

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS 9 6 1 4 3.. 1. 1... 1. 1. 2... 4. 6. 8... 4. 6. 8.. 6 1 4 3. 1. 2. 3.. 1. 2. 3. 4....... 1. a priori LM LM LM.1 Figure 3. Mean absolute difference to true profile for ideal retrievals from bending angles () and refractivity () with different Levenberg Marquardt parameter setting, using 4 iterations. Temperature results (left) and water vapor results (right) for an optimistic a priori (top) and a moderate a priori (bottom).

VON ENGELN AND NEDOLUHA creases at lower altitudes. Bias retrievals show improvement for moderate a priori cases, one Levenberg Marquardt parameter setting even removes the bias completely. Overall, the limited number of processed occultations does not provide a clear picture for the bias of the reference pressure retrieval. As noted above, pressure results are almost independent of the a priori error, thus a smaller a priori error will not improve the presented results. Palmer and Barnett [1] showed that retrieval from GPS/MET observations show generally a negative bias with respect to analysis for surface pressure retrievals, except for tropical occultations. The authors attributed this to the omitting of water vapor above 3 hpa in the retrieval scheme. Water vapor above 3 hpa is not omitted in our retrieval scheme and we find the introduction of a small negative bias in the reference pressure for tropical occultations. Polar occultations generally lead to an improved reference pressure, while no clear picture emerges for mid-latitude occultations.. Realistic Retrieval Case Realistic temperature and water vapor profile retrievals using either bending angle or refractivity measurements were performed for all 1 occultations. The EGOPS tool has been used to simulate the measurement process, including the superposition of realistic errors as discussed above. Retrieval results are presented in Figure 4. Since ray tracer, wave optics, and ideal retrieval have in general different altitude coverage in the troposphere, profiles have been aligned to the lowest altitude covered in all retrieval cases of that particular occultation. Optimistic a priori: Quasi-realistic retrieval of temperature from bending angle profiles results in a mean absolute difference to the true profile of below.2 K between 11 km and 2 km, ray tracer and wave optics results are very similar. The error increases above 2 km caused by the decreasing signal-to-noise ratio. The retrieved temperatures are on average worse than the a priori above about 48 km, caused by noise on the measurement (except for the ideal retrieval case). The a priori data is too accurate over here, thus no improvement is achieved. The impact of water vapor can be seen below about 13 km, where the retrieval s mean absolute error increases. Ray tracer and wave optics calculations yield very similar results at all altitudes. Only small improvements are detectable in the ideal temperature profile retrieval at altitudes below 7 km, ray tracer and wave optics calculations do not lead to an improvement within this altitude range. Ideal retrievals yield better results at all altitudes and are unaffected by noise at higher altitudes, where they still lead to a small improvement of the detected temperatures. The retrieval of temperature from refractivity profiles is very similar to the bending angle results, except for high altitudes where the smoothing introduced by the bending angle to refractivity processing reduces the noise on the measurement and thus yields an improvement in the detected temperatures. Water vapor retrieval from bending angles and refractivity profiles look very similar in Figure 4. Ray tracer results are slightly better than wave optics ones since wave optics calculations will introduce a vertical smoothing of the vertically highly variable water vapor field. Both have problems below about 2 km, where the a priori error is very low. Above, an improvement of a few percent over the a priori is achieved. Ideal retrievals show better results at all altitudes, with improvements of about %. All results look very similar at altitudes above km. Moderate a priori: Temperature retrieval from bending angles yields an improvement in temperature for almost all altitudes (Figure 4). No improvement is achieved only within the lowest few kilometers. The higher a priori difference to the true profile above about 48 km still leads to improvements, contrary to the optimistic case. Hence the optimistic a priori gives a lower limit for temperature retrieval accuracies from bending angle profiles at high altitudes (for the applied measurement error settings within EGOPS). Wave optics, ray tracer, and ideal retrieval give very similar results for all altitudes, indicating that the spherical symmetry assumption used in the ideal case is also valid for the realistic retrievals on average. Retrieving temperature from refractivity profiles shows very similar results to the bending angle retrieval, although slightly better results are obtained in the upper troposphere/lower stratosphere. Retrievals of water vapor yield an improvement over the a priori at almost all altitudes below 6. km. The improvement is only about % to % at lower altitudes, and improves to almost % around 4 km to km. Refractivity and bending angle profiles yield very similar results, except around 7. km, where the above mentioned problems with slow convergence to the minimum of the cost function still occur for retrievals from bending angle profiles. In general, wave optics and ray tracer calculations yield very similar results for both, temperature and water vapor retrieval (see Figure 4). As mentioned in von Engeln et al. [doi:.29/2jd298, 3], the integration of bending angles and refractivity profiles to the applied retrieval grid follows a similar process as the wave optics calculation, thus partly removing the impact of Fresnel diffraction. No canonical transform processing of the measurement, as discussed in Gorbunov [2b], is generally necessary for this retrieval resolution. Although profiles with a lot of sharp structure will show differences between a ray tracer and a wave optics calculation, and canonical transform processing

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS 11 6 1 4 3.. 1. 1... 1. 1. 2.... 3. 4.... 3. 4.. 6 1 4 3. 1. 2. 3.. 1. 2. 3. 4... 4. 6. 8... 4. 6. 8.. a priori RT WO Ideal Figure 4. Mean absolute difference to true profile for realistic and ideal retrievals from bending angles () and refractivity () using a ray tracer (RT), wave optics (WO), or ideal model. Temperature results (left) and water vapor results (right) for an optimistic a priori (top) and a moderate a priori (bottom).

12 VON ENGELN AND NEDOLUHA Table 3. Bias and mean absolute error of pressure retrieval from ideal retrievals for different Levenberg Marquardt parameter setting, optimistic a priori (O), moderate a priori (M). A Priori Retrieved Case Mean Bias Mean Bias O-LM. % -.1 %.12 %. % O-LM. % -.1 %.13 %.3 % O-LM.1. % -.1 %.14 %.2 % M-LM.28 %. %.37 %.6 % M-LM.28 %. %.39 %. % M-LM.1.28 %. %.48 % -.6 % might be necessary. Also, higher retrieval resolutions will show more pronounced differences between a ray tracer and a wave optics simulation. Figure shows the realistic retrieval results from ray tracer simulations, separated by latitude band. The mean latitude θ of an occultation defines the latitude band as: tropical θ < 3, polar θ > 6, and mid-latitudes in between. In total, 46 occultations fall into the tropical band, 38 into the mid-latitude one, and 26 into the polar one. One should keep in mind that our study is focusing on only 1 occultations and that a latitude separation does not give a statistically meaningful analysis with respect to ECMWF data. Optimistic a priori: Retrieval from bending angle and refractivity profiles are very similar, both in temperature and water vapor capabilities, except for tropical and mid-latitude temperature results above about 4 km to 4 km. Here, bending angle retrievals are more affected by the low signal-tonoise ratio and do not yield an improvement over the a priori. No improvement in temperature is achieved in the lower troposphere, especially for tropical and mid-latitude occultations. Water vapor retrievals show improvements over the a priori profile for tropical and mid-latitude occultations at altitudes between about 2 km and 8 km. Very low water vapor levels for polar occultations do not yield any improvement over the a priori profile. Refractivity and bending angle assimilations yield very similar results, although calculations with a higher number of iterations show potential for improved tropical water vapor retrieval from refractivity profiles, all other results are unaffected by an increase in the number of iterations. Moderate a priori: Tropical temperature retrievals are similar to results obtained from an optimistic a priori (Figure ), with 2 exceptions: (1) retrieval from bending angle and refractivity profiles both lead to an improvement over the a priori at higher altitudes. (2) depending on the quality of the a priori, retrieval from refractivity or bending angle profiles show differences. Especially the range from about 1 km to 3 km shows better results for mid-latitude and polar occultations when assimilating refractivity profiles. Tropical retrievals have a lower a priori error and thus yield similar results as the optimistic a priori within this altitude range. Bending angles perform better above about 4 km, especially for tropical occultations. The amount of water vapor determines the temperature retrieval capabilities at lower altitudes, no improvement over the a priori is gained for altitudes below about 9 km (6 km) for tropical (mid-latitude) occultations. Temperature corrections are performed by the 1D Var scheme to compensate for differences in water vapor at altitudes where the sensitivity toward water vapor retrieval is low. This leads to temperature corrections that are worse than the a priori information at about 7 km ( km) for tropical (mid-latitude) occultations. Tropical water vapor retrievals show a substantial improvement over the a priori for altitudes between about 1. km and 8 km. Midlatitude water vapor retrievals improve the water vapor profile below about km, but reveal problems with separating the impact of water vapor and temperature correctly at about 7. km. Only one occultation is responsible for this, removing this occultations yields improvement over the a priori profile up to about 7. km. Polar water vapor retrievals yield only very limit improvements at altitudes below about 3 km. No discernible difference is visible between bending angle and refractivity assimilation. As mentioned above, a priori profiles have to be chosen very carefully in order to avoid the introduction of a bias. Nevertheless, biases will in general be present, thus the possibility to remove biases from an in general biased a priori profile could be another valuable contribution of radio occultation data. Biases in the a priori field could come from biased assimilated data that enters the NWP model, or ap-

TRIEVAL FROM RADIO OCCULTATION MEASUMENTS 13 6 T M P T M P 1 4 3.. 1. 1... 1. 1... 1. 1. 2.... 3. 4.... 3. 4.... 3. 4.. 6 T M P T M P 1 4 3. 1. 2. 3.. 1. 2. 3.. 1. 2. 3. 4.......... 1. a priori RT RT Figure. Mean absolute difference to true profile for realistic ray tracer (RT) simulations from bending angles () and refractivity (), separated by latitude band: tropical (T), mid-latitude (M), polar (P). Temperature results (left) and water vapor results (right) for an optimistic a priori (top) and a moderate a priori (bottom).

14 VON ENGELN AND NEDOLUHA proximations made within the NWP model to calculate the atmospheric behavior. Figure 6 shows the bias removal capabilities from bending angle and refractivity profile retrieval. Optimistic a priori: No major bias is present in the temperature a priori data below 3 km, even though a bias in NWP models that is below. K still can have a severe impact on global climate change studies. Stratospheric temperature biases are in general reduced, while tropospheric retrievals are problematic due to the presence of water vapor. Biases present in the water vapor profiles are removed partly, in particular for tropical occultations between 2 km and km. Hardly any difference is evident between retrieval from bending angle or refractivity profiles, although refractivity shows better results for tropical occultations with an increase in the number of iterations. Moderate a priori: Temperature biases of more than 1 K are present in the a priori data (Figure 6). The retrieval removes most of these biases in the stratosphere, especially for tropical and polar occultations. Mid-latitude occultations show some problems for retrieval from bending angle profiles between about 19 km and 38 km. The refractivity profiles are less affected, because the processing of bending angle to refractivity introduces smoothing of the refractivity profile. Within the troposphere, mid-latitude retrievals produce a strong bias at about km. This bias is also present in ideal retrievals. Looking at the individual retrievals shows that temperature adjustments are performed to compensate for water vapor differences between the true and the a priori profile, at an altitude where the sensitivity of the retrieval toward water vapor is very low. This indicates that severely wrong water vapor a priori can hamper temperature bias removal. The retrieval of water vapor removes part of the bias present in the tropical a priori data. Mid-latitude and polar occultations show a reduction in the bias below about 3 km to 4 km, although the number of retrievals entering the bias calculation decreases at lower altitude. Thus a larger ensemble has to be processed for more reliable results. Contrary to the mid-latitude temperature results, water vapor shows only a small bias around km. Further analysis reveals that the retrieval has a tendency to compensate only too low a priori water vapor profiles by varying the temperature, thus leading to a bias in the temperature results. The bias is also observed using a linear scale for water vapor VMR retrieval instead of the logarithmic one, thus it cannot be attributed to the assumed shape of the Gaussian distribution on a linear or a logarithmic scale. The introduced retrieval bias at midlatitudes around 7. km is caused by the already mentioned problematic occultation. Removing this occultation from the calculations removes the present bias. The magnitude of the bias is reduced by a linear VMR retrieval scale. 6. Impact of A Priori Errors A priori covariance matrix errors S have been varied by factors of. and 2. separately for temperature and water vapor to determine the impact of these errors on the retrieval accuracy. A higher a priori covariance matrix error will put more weight on the measurement, while lower errors will lead to more restrained retrievals. Results for temperature a priori covariance matrix error variations using realistic retrievals from bending angle and refractivity profiles are shown in Figure 7. Optimistic a priori: Temperature retrieval from bending angle profiles at altitudes above about 48 km and below about 8 km requires stronger constraints. Noise is responsible for the deviations at upper altitudes, while the quality of the water vapor retrieval affects the retrieval at lower altitudes. Temperature retrieval from refractivity profiles is not affected by noise at higher altitude (caused by the smoothing, as mentioned above), but suffers from the same problems at lower altitudes. Improvements in the temperature retrieval are possible with a less constraint a priori covariance matrix error between about 2 km and 48 km, where retrieval from refractivity profiles shows larger improvements. Variation factors of 1. and 2. do not differ significantly, even though small improvements are detectable with a less constraint a priori at altitudes between about 39 km and 47 km for retrieval from refractivity profiles. Water vapor retrievals look very similar for bending angle and refractivity profile assimilation. Low temperature a priori covariance matrix errors will in general lead to improved water vapor results for altitudes above about km, since the temperature retrieval is more constraint and erroneous retrievals as discussed in Section 4 are less likely to occur. The picture below km is ambiguous. Tighter constraints on the temperature a priori data will only allow small adjustments in temperature. There are a priori temperature profiles that have larger deviations than given by the a priori errors, thus required adjustments in temperature will be restricted by the tighter a priori. Water vapor adjustments have to be substantial to compensate for these temperature deviations, hence the individual profiles will show up in the averaged water vapor data. Moderate a priori: Temperature retrievals from bending angle and refractivity profiles both show improvements at higher altitude with a less constrained a priori covariance matrix error, the noise problems present in the optimistic a priori are almost completely removed. Retrieval from refractivity profiles show larger improvements at higher altitudes for higher a priori covariance matrix errors, indicating that retrieval from refractivity profiles has advantages over bending angle profiles at high altitudes. Even though the signalto-noise ratio of the bending angle and refractivity measure-